Connections on modules over singularities

Researchers

• Eivind Eriksen, Eivind.Eriksen at bi.no
• Trond Stølen Gustavsen, Trond.Gustavsen at usn.no

Papers

1. Equivariant Lie-Rinehart cohomology (Eriksen, Gustavsen) - [Proc. Estonian Acad. Sci. Phys. Math.
2. Lie-Rinehart cohomology and integrable connections on modules of rank one (Eriksen, Gustavsen) - [J. Algebra 322 (2009), no. 12, 4283-4294]
3. Connections on modules over singularities of finite and tame CM representation type (Eriksen, Gustavsen) - [Generalized Lie Theory in Mathematics, Physics and Beyond, 99-108, Springer, 2008]
4. Connections on modules over quasi-homogeneous plane curves (Eriksen) - [Comm. Algebra, 36 (2008), no. 8, 3032 - 3041]
5. Connections on modules over singularities of finite CM representation type (Eriksen, Gustavsen) - [J. Pure Appl. Algebra 212 (2008), no. 7, pp. 1561-1574] / Extended preprint version
6. Computing obstructions for existence of connections on modules (Eriksen, Gustavsen) - [J. Symb. Comput. 42 (2007), no. 3, pp. 313-323]

Libraries

• The Singular library connections.lib (Version 1.24, 26/05 2008) for calculations of connections on modules.
• News in v1.24: Name of library changed to connections.lib, new procedures for curvature/integrability, minor corrections. Internal procedures no longer static.
  
• News in v1.20: New procedure Conn to compute a (V-) connection of a module, some minor corrections.
• News in v1.12: Only minor changes of notation
• News in v1.11: Only minor changes of notation
• News in v1.10: Corrected and updated version of the procedure AClass(formerly called KSClass), which calculates the Atiyah class of a module.

Example scripts

• Simple zero-dimensional singularities:
1. The singularity An: simple-an-dim0.sing
• Simple curve singularities:
1. The singularity An: simple-aneven-dim1.sing - simple-anodd-dim1.sing
   
2. The singularity Dn: simple-dnodd-dim1.sing - simple-dneven-dim1.sing
   
3. The singularity E6: simple-e6-dim1.sing
4. The singularity E7: simple-e7-dim1.sing
5. The singularity E8: simple-e8-dim1.sing
• Non-Gorenstein curve singularities of finite CM type::
1. The singularity k[[t³,t⁴,t⁵]]: monomial-345.sing
2. The singularity k[[t³,t⁵,t⁷]]: monomial-357.sing
3. The singularity k[[x,y,z]]/(x²-yⁿ,xz,yz): dsnodd-dim1.sing
   
4. The singularity k[[x,y,z]]/(x³-y⁴,xz-y², y²z-x²,yz²-xy): e7s-dim1.sing
5. The singularity k[[x,y,z]]/(x²-xy^n/2,xz,yz): dsneven-dim1.sing
   
• Simple surface singularities:
1. The singularity An: simple-an-dim2.sing
   
• Simple threefold singularities:
1. The singularity An: simple-aneven-dim3.sing - simple-anodd-dim3.sing
   
2. The singularity Dn: simple-dnodd-dim3.sing - simple-dneven-dim3.sing
   
3. The singularity E6: simple-e6-dim3.sing
4. The singularity E7: simple-e7-dim3.sing
5. The singularity E8: simple-e8-dim3.sing
• Non-Gorenstein threefold singularities of finite CM type:
  
1. The threefold quotient singularityof finite CM type: quotient-dim3.sing
2. The threefold scroll of type (1,2): scroll-21-dim3.sing
• Simple fourfold singularities:
1. The singularity An: simple-an-dim4.sing
• Some singularities which are not of finite CM type:
1. The monomical curve singularity k[[t⁴,t⁵,t⁶,t⁷]]: monomial-4567.sing
2. The cubic surface singularity k[[x,y,z]]/(x³ + y³ + z³): cubic-dim2.sing

• An example of a connection with non-zero curvature:
1. A non-graded connection on the cubic surface singularity k[[x,y,z]]/(x³ + y³ + z³): cubic-curv.sing